Operasi Aljabar Pada Bentuk Akar 1. 2. 3. 4. 5. Soal di atas dapat diselesaikan dengan Jadi, hasil dari adalah .

Hasil dari ....

Rasionalkan penyebut pecahan-pecahan berikut! e. f. g. h.

Pembahasan Soal:

e. Diketahui $fraction numerator 5 over denominator square root of 12 minus 3 end fraction$ maka cara untuk merasionalkannya adalah dengan mengalikan akar sekawan dari $square root of 12 minus 3$ yaitu $square root of 12 plus 3$ sehingga diperoleh

Berdasarkan aturan perkalian bentuk akar dengan akar pangkat sama yaitu $n-th root of a cross times n-th root of b equals n-th root of a cross times b end root$ maka

$table attributes columnalign right center left columnspacing 0px end attributes row cell fraction numerator 5 over denominator square root of 12 minus 3 end fraction end cell equals cell fraction numerator 5 over denominator square root of 12 minus 3 end fraction cross times fraction numerator square root of 12 plus 3 over denominator square root of 12 plus 3 end fraction end cell row blank equals cell fraction numerator 5 open parentheses square root of 12 plus 3 close parentheses over denominator open parentheses square root of 12 minus 3 close parentheses open parentheses square root of 12 plus 3 close parentheses end fraction end cell row blank equals cell fraction numerator 5 cross times square root of 12 plus 5 cross times 3 over denominator square root of 12 cross times square root of 12 plus 3 cross times square root of 12 minus 3 cross times square root of 12 minus 3 cross times 3 end fraction end cell row blank equals cell fraction numerator 5 square root of 12 plus 15 over denominator square root of 12 cross times 12 end root plus up diagonal strike 3 square root of 12 minus 3 square root of 12 end strike minus 9 end fraction end cell row blank equals cell fraction numerator 5 square root of 12 plus 15 over denominator square root of 12 squared end root minus 9 end fraction end cell row blank equals cell fraction numerator 5 square root of 12 plus 15 over denominator 12 minus 9 end fraction end cell row blank equals cell fraction numerator 5 square root of 12 plus 15 over denominator 3 end fraction end cell end table$

Jika $n-th root of a$ dapat difaktorkan menjadi $n-th root of p cross times q end root$ dengan $p space text atau end text space q$ bilangan berpangkat $n$ maka $n-th root of a$ dapat disederhanakan sehingga

Jadi bentuk rasional dari $fraction numerator 5 over denominator square root of 12 minus 3 end fraction$ adalah $table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell fraction numerator 10 square root of 3 plus 15 over denominator 3 end fraction end cell end table$ .

f. Diketahui $fraction numerator 6 square root of 3 over denominator square root of 7 plus square root of 6 end fraction$ maka cara merasionalkannya adalah dengan mengalikan penyebut akar dengan akar sekawannya yaitu $square root of 7 minus square root of 6$ sehingga

Berdasarkan aturan perkalian bentuk akar dengan akar pangkat sama yaitu $n-th root of a cross times n-th root of b equals n-th root of a cross times b end root$ maka

$table attributes columnalign right center left columnspacing 0px end attributes row cell fraction numerator 6 square root of 3 over denominator square root of 7 plus square root of 6 end fraction end cell equals cell fraction numerator 6 square root of 3 over denominator square root of 7 plus square root of 6 end fraction cross times fraction numerator square root of 7 minus square root of 6 over denominator square root of 7 minus square root of 6 end fraction end cell row blank equals cell fraction numerator 6 square root of 3 open parentheses square root of 7 minus square root of 6 close parentheses over denominator open parentheses square root of 7 plus square root of 6 close parentheses open parentheses square root of 7 minus square root of 6 close parentheses end fraction end cell row blank equals cell fraction numerator 6 square root of 3 cross times square root of 7 minus 6 square root of 3 cross times square root of 6 over denominator square root of 7 cross times square root of 7 minus square root of 7 cross times square root of 6 plus square root of 6 cross times square root of 7 minus square root of 6 cross times square root of 6 end fraction end cell row blank equals cell fraction numerator 6 square root of 3 cross times 7 end root minus 6 square root of 3 cross times 6 end root over denominator square root of 7 cross times 7 end root minus square root of 7 cross times 6 end root plus square root of 6 cross times 7 end root minus square root of 6 cross times 6 end root end fraction end cell row blank equals cell fraction numerator 6 square root of 21 minus 6 square root of 18 over denominator square root of 49 up diagonal strike negative square root of 42 plus square root of 42 end strike minus square root of 36 end fraction end cell row blank equals cell fraction numerator 6 square root of 21 minus 6 square root of 18 over denominator square root of 49 minus square root of 36 end fraction end cell end table$

Jadi bentuk rasional dari $fraction numerator 6 square root of 3 over denominator square root of 7 plus square root of 6 end fraction$ adalah $table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell 6 square root of 21 minus 18 square root of 2 end cell end table$ .

g. Diketahui $fraction numerator 4 plus square root of 5 over denominator square root of 15 minus square root of 8 end fraction$ maka cara merasionalkannya adalah dengan mengalikan penyebut dengan akar sekawannya yaitu $square root of 15 plus square root of 8$ sehingga

Berdasarkan aturan perkalian bentuk akar dengan akar pangkat sama yaitu $n-th root of a cross times n-th root of b equals n-th root of a cross times b end root$ maka

$table attributes columnalign right center left columnspacing 0px end attributes row cell fraction numerator 4 plus square root of 5 over denominator square root of 15 minus square root of 8 end fraction end cell equals cell fraction numerator 4 plus square root of 5 over denominator square root of 15 minus square root of 8 end fraction cross times fraction numerator square root of 15 plus square root of 8 over denominator square root of 15 plus square root of 8 end fraction end cell row blank equals cell fraction numerator open parentheses 4 plus square root of 5 close parentheses open parentheses square root of 15 plus square root of 8 close parentheses over denominator open parentheses square root of 15 minus square root of 8 close parentheses open parentheses square root of 15 plus square root of 8 close parentheses end fraction end cell row blank equals cell fraction numerator 4 cross times square root of 15 plus 4 cross times square root of 8 plus square root of 5 cross times square root of 15 plus square root of 5 cross times square root of 8 over denominator square root of 15 cross times square root of 15 plus square root of 15 cross times square root of 8 minus square root of 8 cross times square root of 15 minus square root of 8 cross times square root of 8 end fraction end cell row blank equals cell fraction numerator 4 square root of 15 plus 4 square root of 8 plus square root of 5 cross times 15 end root plus square root of 5 cross times 8 end root over denominator square root of 15 cross times 15 end root plus square root of 15 cross times 8 end root minus square root of 8 cross times 15 end root minus square root of 8 cross times 8 end root end fraction end cell row blank equals cell fraction numerator 4 square root of 15 plus 4 square root of 8 plus square root of 75 plus square root of 40 over denominator square root of 15 squared end root plus square root of 120 minus square root of 120 minus square root of 64 end fraction end cell end table$

Jadi bentuk rasional dari $fraction numerator 4 plus square root of 5 over denominator square root of 15 minus square root of 8 end fraction$ adalah $table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell fraction numerator 4 square root of 15 plus 4 square root of 8 plus 5 square root of 3 plus 2 square root of 10 over denominator 7 end fraction end cell end table$ .

h. Diketahui $fraction numerator square root of 8 minus square root of 2 over denominator 4 square root of 3 plus square root of 6 end fraction$ maka cara merasionalkannya adalah dengan mengalikan penyebut dengan akar sekawannya yaitu $4 square root of 3 minus square root of 6$ sehingga

Berdasarkan aturan perkalian bentuk akar dengan akar pangkat sama yaitu $n-th root of a cross times n-th root of b equals n-th root of a cross times b end root$ maka

$table attributes columnalign right center left columnspacing 0px end attributes row cell fraction numerator square root of 8 minus square root of 2 over denominator 4 square root of 3 plus square root of 6 end fraction end cell equals cell fraction numerator square root of 8 minus square root of 2 over denominator 4 square root of 3 plus square root of 6 end fraction cross times fraction numerator 4 square root of 3 minus square root of 6 over denominator 4 square root of 3 minus square root of 6 end fraction end cell row blank equals cell fraction numerator open parentheses square root of 8 minus square root of 2 close parentheses open parentheses 4 square root of 3 minus square root of 6 close parentheses over denominator open parentheses 4 square root of 3 plus square root of 6 close parentheses open parentheses 4 square root of 3 minus square root of 6 close parentheses end fraction end cell row blank equals cell fraction numerator square root of 8 cross times 4 square root of 3 minus square root of 8 cross times square root of 6 minus square root of 2 cross times 4 square root of 3 plus square root of 2 cross times square root of 6 over denominator 4 square root of 3 cross times 4 square root of 3 minus 4 square root of 3 cross times square root of 6 plus square root of 6 cross times 4 square root of 3 minus square root of 6 cross times square root of 6 end fraction end cell row blank equals cell fraction numerator 4 square root of 8 cross times 3 end root minus square root of 8 cross times 6 end root minus 4 square root of 2 cross times 3 end root plus square root of 2 cross times 6 end root over denominator 16 square root of 3 cross times 3 end root minus 4 square root of 3 cross times 6 end root plus 4 square root of 6 cross times 3 end root minus square root of 6 cross times 6 end root end fraction end cell row blank equals cell fraction numerator 4 square root of 24 minus square root of 48 minus 4 square root of 6 plus square root of 12 over denominator 16 square root of 9 minus 4 square root of 18 plus 4 square root of 18 minus square root of 36 end fraction end cell row blank equals cell fraction numerator 4 square root of 4 cross times 6 end root minus square root of 16 cross times 3 end root minus 4 square root of 6 plus square root of 4 cross times 3 end root over denominator 16 cross times 9 minus 6 end fraction end cell row blank equals cell fraction numerator 4 square root of 2 squared cross times 6 end root minus square root of 4 squared cross times 3 end root minus 4 square root of 6 plus square root of 2 squared cross times 3 end root over denominator 144 minus 6 end fraction end cell row blank equals cell fraction numerator 4 cross times 2 square root of 6 minus 4 square root of 3 minus 4 square root of 6 plus 2 square root of 3 over denominator 138 end fraction end cell row blank equals cell fraction numerator 8 square root of 6 minus 4 square root of 3 minus 4 square root of 6 plus 2 square root of 3 over denominator 138 end fraction end cell end table$

Berdasarkan aturan penjumlahan dan pengurangan bentuk akar dengan bilangan pokok sama yaitu $a n-th root of b plus-or-minus c n-th root of b equals open parentheses a plus-or-minus c close parentheses n-th root of b$ maka

Jadi bentuk rasional dari $fraction numerator square root of 8 minus square root of 2 over denominator 4 square root of 3 plus square root of 6 end fraction$ adalah $table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell fraction numerator 4 square root of 6 minus 2 square root of 3 over denominator 138 end fraction end cell end table$ .

Roboguru

Tentukan hasil operasi bilangan berikut! d. e. f. g.

Pembahasan Soal:

Ingat sifat distributif perkalian sebagai berikut.

$table attributes columnalign right center left columnspacing 0px end attributes row cell a open parentheses b plus c close parentheses end cell equals cell a b plus b c end cell row cell open parentheses a plus b close parentheses open parentheses c plus d close parentheses end cell equals cell a c plus a d plus b c plus b d end cell end table$

Sehingga

d. $square root of 3 open parentheses 2 square root of 6 minus square root of 25 close parentheses$ diperoleh sebagai berikut.

Berdasarkan aturan perkalian bentuk akar dengan akar pangkat sama yaitu $n-th root of a cross times n-th root of b equals n-th root of a cross times b end root comma a not equal to 0 space text dan end text space b not equal to 0$ maka

$table attributes columnalign right center left columnspacing 0px end attributes row cell square root of 3 open parentheses 2 square root of 6 minus square root of 25 close parentheses end cell equals cell square root of 3 cross times 2 square root of 6 minus square root of 3 cross times square root of 25 end cell row blank equals cell square root of 3 cross times 2 square root of 6 minus square root of 3 cross times 5 end cell row blank equals cell 2 square root of 3 cross times 6 end root minus 5 square root of 3 end cell row blank equals cell 2 square root of 18 minus 5 square root of 3 end cell row blank blank blank end table$

$table attributes columnalign right center left columnspacing 0px end attributes row cell square root of 3 open parentheses 2 square root of 6 minus square root of 25 close parentheses end cell equals cell 2 square root of 3 cross times 6 end root minus 5 square root of 3 end cell row blank equals cell 2 square root of 18 minus 5 square root of 3 end cell row blank equals cell 2 square root of 9 cross times 2 end root minus 5 square root of 3 end cell row blank equals cell 2 square root of 3 squared cross times 2 end root minus 5 square root of 3 end cell row blank equals cell 2 cross times 3 square root of 2 minus 5 square root of 3 end cell row blank equals cell 6 square root of 2 minus 5 square root of 3 end cell end table$

Jadi hasil dari $square root of 3 open parentheses 2 square root of 6 minus square root of 25 close parentheses$ adalah $table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell 6 square root of 2 minus 5 square root of 3 end cell end table$ .

e. $open parentheses square root of 6 plus square root of 8 close parentheses open parentheses square root of 6 minus square root of 3 close parentheses$ Diperoleh sebagai berikut.

$begin mathsize 12px style open parentheses square root of 6 plus square root of 8 close parentheses open parentheses square root of 6 minus square root of 3 close parentheses equals square root of 6 cross times square root of 6 minus square root of 6 cross times square root of 3 plus square root of 8 cross times square root of 6 minus square root of 8 cross times square root of 3 end style$

Berdasarkan aturan perkalian bentuk akar dengan akar pangkat sama yaitu $n-th root of a cross times n-th root of b equals n-th root of a cross times b end root$ maka

$begin mathsize 12px style table attributes columnalign right center left columnspacing 0px end attributes row cell open parentheses square root of 6 plus square root of 8 close parentheses open parentheses square root of 6 minus square root of 3 close parentheses end cell equals cell square root of 6 cross times square root of 6 minus square root of 6 cross times square root of 3 plus square root of 8 cross times square root of 6 minus square root of 8 cross times square root of 3 end cell row blank equals cell square root of 6 cross times 6 end root minus square root of 6 cross times 3 end root plus square root of 8 cross times 6 end root minus square root of 8 cross times 3 end root end cell row blank equals cell square root of 36 minus square root of 18 plus square root of 48 minus square root of 24 end cell end table end style$

$table attributes columnalign right center left columnspacing 0px end attributes row cell open parentheses square root of 6 plus square root of 8 close parentheses open parentheses square root of 6 minus square root of 3 close parentheses end cell equals cell square root of 6 cross times 6 end root minus square root of 6 cross times 3 end root plus square root of 8 cross times 6 end root minus square root of 8 cross times 3 end root end cell row blank equals cell square root of 36 minus square root of 18 plus square root of 48 minus square root of 24 end cell row blank equals cell 6 minus square root of 9 cross times 2 end root plus square root of 16 cross times 3 end root minus square root of 4 cross times 6 end root end cell row blank equals cell 6 minus square root of 3 squared cross times 3 end root plus square root of 4 squared cross times 3 end root minus square root of 2 squared cross times 6 end root end cell row blank equals cell 6 minus 3 square root of 3 plus 4 square root of 3 minus 2 square root of 6 end cell end table$

Bentuk akar dengan akar dan bilangan pokok sama dapat dikurangkan dengan menggunakan sifat distributif sebagai berikut

$a n-th root of b minus c n-th root of b equals open parentheses a minus c close parentheses n-th root of b$

Sehingga

Jadi hasil dari $open parentheses square root of 6 plus square root of 8 close parentheses open parentheses square root of 6 minus square root of 3 close parentheses$ adalah $table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell 6 plus square root of 3 minus 2 square root of 6 end cell row blank blank blank end table$ .

f. $open parentheses 2 square root of 3 minus square root of 6 close parentheses open parentheses square root of 8 minus square root of 2 close parentheses$ diperoleh sebagai berikut.

$begin mathsize 12px style open parentheses 2 square root of 3 minus square root of 6 close parentheses open parentheses square root of 8 minus square root of 2 close parentheses equals 2 square root of 3 cross times square root of 8 minus 2 square root of 3 cross times square root of 2 minus square root of 6 cross times square root of 8 plus square root of 6 cross times square root of 2 end style$

Berdasarkan perkalian bentuk akar dengan akar pangkat sama yaitu $n-th root of a cross times n-th root of b equals n-th root of a cross times b end root$ maka

$begin mathsize 12px style table attributes columnalign right center left columnspacing 0px end attributes row cell open parentheses 2 square root of 3 minus square root of 6 close parentheses open parentheses square root of 8 minus square root of 2 close parentheses end cell equals cell 2 square root of 3 cross times square root of 8 minus 2 square root of 3 cross times square root of 2 minus square root of 6 cross times square root of 8 plus square root of 6 cross times square root of 2 end cell row blank equals cell 2 square root of 3 cross times 8 end root minus 2 square root of 3 cross times 2 end root minus square root of 6 cross times 8 end root plus square root of 6 cross times 2 end root end cell row blank equals cell 2 square root of 24 minus 2 square root of 6 minus square root of 48 plus square root of 12 end cell row blank blank blank end table end style$

$table attributes columnalign right center left columnspacing 0px end attributes row cell open parentheses 2 square root of 3 minus square root of 6 close parentheses open parentheses square root of 8 minus square root of 2 close parentheses end cell equals cell 2 square root of 3 cross times 8 end root minus 2 square root of 3 cross times 2 end root minus square root of 6 cross times 8 end root plus square root of 6 cross times 2 end root end cell row blank equals cell 2 square root of 24 minus 2 square root of 6 minus square root of 48 plus square root of 12 end cell row blank equals cell 2 square root of 4 cross times 6 end root minus 2 square root of 6 minus square root of 16 cross times 3 end root plus square root of 4 cross times 3 end root end cell row blank equals cell 2 square root of 2 squared cross times 6 end root minus 2 square root of 6 minus square root of 4 squared cross times 3 end root plus square root of 2 squared cross times 3 end root end cell row blank equals cell 2 cross times 2 square root of 6 minus 2 square root of 6 minus 4 square root of 3 plus 2 square root of 3 end cell row blank equals cell 4 square root of 6 minus 2 square root of 6 minus 4 square root of 3 plus 2 square root of 3 end cell end table$

Bentuk akar dengan akar dan bilangan pokok sama dapat dikurangkan dengan menggunakan sifat distributif sebagai berikut

$a n-th root of b minus c n-th root of b equals open parentheses a minus c close parentheses n-th root of b$

Sehingga

Jadi hasil dari $open parentheses 2 square root of 3 minus square root of 6 close parentheses open parentheses square root of 8 minus square root of 2 close parentheses$ adalah $table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell 2 square root of 6 minus 2 square root of 3 end cell end table$ .

g. Ingat rumus perkalian suku dua bentuk aljabar yaitu

$open parentheses a plus b close parentheses open parentheses a minus b close parentheses equals a squared minus b squared$

Sehingga $open parentheses 3 square root of 5 minus square root of 6 close parentheses open parentheses 3 square root of 5 plus square root of 6 close parentheses$ diperoleh sebagai berikut.

$open parentheses 3 square root of 5 minus square root of 6 close parentheses open parentheses 3 square root of 5 plus square root of 6 close parentheses equals open parentheses 3 square root of 5 close parentheses squared minus open parentheses square root of 6 close parentheses squared$

Bentuk akar dapat diubah menjadi bilangan berpangkat yaitu $n-th root of a to the power of m end root equals a to the power of m over n end exponent comma space a not equal to 0$ sehingga

Berdasarkan sifat bilangan berpangkat $open parentheses a cross times b close parentheses to the power of m equals a to the power of m cross times b to the power of m$ maka

Berdasarkan sifat bilangan berpangkat $open parentheses a to the power of m close parentheses to the power of n equals a to the power of m cross times n end exponent$ dan $a to the power of 1 equals a$ maka

$table attributes columnalign right center left columnspacing 0px end attributes row cell open parentheses 3 square root of 5 minus square root of 6 close parentheses open parentheses 3 square root of 5 plus square root of 6 close parentheses end cell equals cell open parentheses 3 square root of 5 close parentheses squared minus open parentheses square root of 6 close parentheses squared end cell row blank equals cell open parentheses 3 cross times 5 to the power of 1 half end exponent close parentheses squared minus open parentheses 6 to the power of 1 half end exponent close parentheses squared end cell row blank equals cell 3 squared cross times open parentheses 5 to the power of 1 half end exponent close parentheses squared minus open parentheses 6 to the power of 1 half end exponent close parentheses squared end cell row blank equals cell 9 cross times 5 to the power of 1 half cross times 2 end exponent minus 6 to the power of 1 half cross times 2 end exponent end cell row blank equals cell 9 cross times 5 to the power of 1 minus 6 to the power of 1 end cell row blank equals cell 9 cross times 5 minus 6 end cell row blank equals cell 45 minus 6 end cell row blank equals 39 end table$

Jadi hasil dari $open parentheses 3 square root of 5 minus square root of 6 close parentheses open parentheses 3 square root of 5 plus square root of 6 close parentheses$ adalah $39$ .

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